Question

Tom's, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom's, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom's, Inc., can purchase up to 285 pounds of whole tomatoes, 140 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom's, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom's contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa.

Letting

W = jars of Western Foods Salsa

M = jars of Mexico City Salsa

leads to the formulation (units for constraints are ounces):

Max 1W + 1.25M

s.t.

5W + 7M ≤ 4,560 oz of whole tomatoes

3W + 1M ≤ 2,240 oz of tomato sauce

2W + 2M ≤ 1,600 oz of tomato paste

W, M ≥ 0

The computer solution is shown below.

Optimal Objective Value = 870.00000 Variable Value Reduced Cost W 520.00000 0.00000 M 280.00000 0.00000

Constraint Slack/Surplus Dual Value 1 0.00000 0.12500 2 400.00000 0.00000 3 0.00000 0.18750

Variable Objective Coefficient Allowable Increase Allowable Decrease W 1.00000 0.25000 0.10714 M 1.25000 0.15000 0.25000

Constraint RHS Value Allowable Increase Allowable Decrease 1 4560.00000 1040.00000 400.00000 2 2240.00000 Infinite 400.00000 3 1600.00000 100.00000 297.14286

(a) What is the optimal solution, and what are the optimal production quantities? W 5 Incorrect: Your answer is incorrect. jars M 7 Incorrect: Your answer is incorrect. jars profit $ 860 Incorrect: Your answer is incorrect.

(b) Specify the objective function ranges. (Round your answers to five decimal places.) Western Foods Salsa to Mexico City Salsa to

(c) What are the dual values for each constraint? Interpret each. constraint 1 One additional ounce of whole tomatoes will improve profits by $0.125. One additional ounce of whole tomatoes will improve profits by $400.00. One additional ounce of whole tomatoes will improve profits by $0.188. Additional ounces of whole tomatoes will not improve profits. constraint 2 One additional ounce of tomato sauce will improve profits by $0.125. One additional ounce of tomato sauce will improve profits by $400.00. One additional ounce of tomato sauce will improve profits by $0.188. Additional ounces of tomato sauce will not improve profits. constraint 3 One additional ounce of tomato paste will improve profits by $0.125. One additional ounce of tomato paste will improve profits by $400.00. One additional ounce of tomato paste will improve profits by $0.188. Additional ounces of tomato paste will not improve profits.

(d) Identify each of the right-hand-side ranges. (Round your answers to two decimal places. If there is no upper or lower limit, enter NO LIMIT.)

constraint 1 _____ to _____

constraint 2 _____ to _____

constraint 3 _____to _____

Answer 1

(a)

Optimal solution:

W = 520
M = 280
Profit = $860

(b)

Objective function ranges:

Western Foods Salsa (1 - 0.10714) to (1 + 0.25) i.e. 0.89286 to 1.25000
Mexico City Salsa (1.25 - 0.25) to (1.25 + 0.15) i.e. 1.00000 to 1.40000

(c)

Constraint-1: One additional ounce of whole tomatoes will improve profits by $0.125

Constraint-2: Additional ounces of tomato sauce will not improve profits

Constraint-3: One additional ounce of tomato paste will improve profits by $0.188

(d)

Constraint-1: (4560 - 400) to (4560 + 1040) i.e. 4160 to 5600

Constraint-2: (2240 - 400) to (2240 + inf) i.e. 1840 to NO LIMIT

Constraint-3: (1600 - 297.14) to (1600 + 100) i.e. 1302.86 to 1700